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Universality of single spin asymmetries in hard processes

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### Universality of single spin asymmetries in hard processes

April 20-24, 2006

Cedran Bomhof and Piet Mulders

Content

Universality of Single Spin Asymmetries (SSA) in hard processes

- Introduction
- SSA and time reversal invariance
- Transverse momentum dependence (TMD)

Through TMD distribution and fragmentation functions totransverse momentsandgluonic poles

- Electroweak processes (SIDIS, Drell-Yan and annihilation)
- Hadron-hadron scattering processes
- Gluonic pole cross sections
- Conclusions

Introduction: partonic structure of hadrons

For (semi-)inclusive measurements, cross sections in hard scattering processes factorize into a hard squared amplitude and distribution and fragmentation functions entering in forward matrix elements of nonlocal combinations of quark and gluon field operators (f y or G)

lightcone

TMD

lightfront

FF

Introduction: partonic structure of hadrons

- Quark distribution functions (DF) and fragmentation functions (FF)
- unpolarized

q(x) = f1q(x) and D(z) = D1(z)

- Polarization/polarimetry

Dq(x) = g1q(x) and dq(x) = h1q(x)

- Azimuthal asymmetries

g1T(x,pT) and h1L(x,pT)

- Single spin asymmetries

h1(x,pT) and f1T(x,pT); H1(z,kT) and D1T(z,kT)

- Form factors
- Generalized parton distributions

FORWARD

matrix elements

x section

one hadron in inclusive or semi-inclusive scattering

NONLOCAL

lightcone

NONLOCAL

lightfront

OFF-FORWARD

Amplitude

Exclusive

LOCAL

NONLOCAL

lightcone

SSA and time reversal invariance

- QCD is invariant under time reversal (T)
- Single spin asymmetries (SSA) are T-odd observables, but they are not forbidden!
- For distribution functions a simple distinction between T-even and T-odd DF’s can be made
- Plane wave states (DF) are T-invariant
- Operator combinations can be classified according to their T-behavior (T-even or T-odd)
- Single spin asymmetries involve an odd number of (i.e. at least one) T-odd function(s)
- The hard process at tree-level is T-even; higher order as is required to get T-odd contributions

K1

df

K2

pp-scattering

Intrinsic transverse momenta- In a hard process one probes partons (quarks and gluons)
- Momenta fixed by kinematics (external momenta)

DISx = xB = Q2/2P.q

SIDIS z = zh = P.Kh/P.q

- Also possible for transverse momenta

SIDIS qT = kT – pT

= q + xBP – Kh/zh-Kh/zh

2-particle inclusive hadron-hadron scattering

qT = p1T + p2T – k1T – k2T

= K1/z1+ K2/z2- x1P1- x2P2 K1/z1+ K2/z2

- Sensitivity for transverse momenta requires 3 momenta

SIDIS: g* + H h + X

DY: H1 + H2 g* + X

e+e-: g* h1 + h2 + X

hadronproduction: H1 + H2 h + X

h1 + h2 + X

p x P + pT

k z-1 K + kT

In azimuthal asymmetries

Transverse moment

TMD correlation functions (unpolarized hadrons)quark correlator

F(x, pT)

- T-odd
- Transversely
- polarized quarks

Color gauge invariance

- Nonlocal combinations of colored fields must be joined by a gauge link:
- Gauge link structure is calculated from collinear A.n gluons exchanged between soft and hard part
- Link structure for TMD functions

depends on the hard process!

DIS F[U]

SIDIS F[U+] =F[+]

DY F[U-] = F[-]

Gluonic poles

- Thus

F[±]a(x) = Fa(x) + CG[±]pFGa(x,x)

- CG[±] = ±1
- with universal functions in gluonic pole m.e. (T-odd for distributions)
- There is only one function h1(1)(x) [Boer-Mulders] and (for transversely polarized hadrons) only onefunction f1T(1)(x) [Sivers] contained in pFG
- These functions appear with a process-dependent sign
- Situation for FF is more complicated because there are no T constraints

What about other hard processes (in particular pp scattering)?

Efremov and Teryaev 1982; Qiu and Sterman 1991

Boer, Mulders, Pijlman, NPB 667 (2003) 201

C. Bomhof, P.J. Mulders and F. Pijlman, PLB 596 (2004) 277

Link structure for fields in correlator 1

Other hard processes- qq-scattering as hard subprocess
- insertions of gluons collinear with parton 1 are possible at many places
- this leads for ‘external’ parton fields to a gauge link to lightcone infinity

C. Bomhof, P.J. Mulders and F. Pijlman, PLB 596 (2004) 277

Other hard processes- qq-scattering as hard subprocess
- insertions of gluons collinear with parton 1 are possible at many places
- this leads for ‘external’ parton fields to a gauge link to lightcone infinity
- The correlator F(x,pT) enters for each contributing term in squared amplitude with specific link

U□ = U+U-†

F[Tr(U□)U+](x,pT)

F[U□U+](x,pT)

Gluonic pole cross sections

- Thus

F[U]a(x) = Fa(x) + CG[U]pFGa(x,x)

- CG[U±] = ±1

CG[U□U+] = 3, CG[Tr(U□)U+] = Nc

- with the same uniquely defined functions in gluonic pole m.e. (T-odd for distributions)

Bacchetta, Bomhof, Pijlman, Mulders, PRD 72 (2005) 034030; hep-ph/0505268

D1

CG [D1]

= CG [D2]

D2

D3

CG [D3]

= CG [D4]

D4

examples: qqqqy

Gluonic pole cross sections- In order to absorb the factors CG[U], one can define specific hard cross sections for gluonic poles (to be used with functions in transverse moments)
- for pp:

etc.

- for SIDIS:

for DY:

- Similarly for gluon processes

Bomhof, Mulders, Pijlman, EPJ; hep-ph/0601171

Conclusions

- Single spin asymmetries in hard processes can exist
- They are T-odd observables, which can be described in terms of T-odd distribution and fragmentation functions
- For distribution functions the T-odd functions appear in gluonic pole matrix elements
- Gluonic pole matrix elements are part of the transverse moments appearing in azimuthal asymmetries
- Their strength is related to path of color gauge link in TMD DFs which may differ per term contributing to the hard process
- The gluonic pole contributions can be written as a folding of universal (soft) DF/FF and gluonic pole cross sections

Belitsky, Ji, Yuan, NPB 656 (2003) 165

Boer, Mulders, Pijlman, NPB 667 (2003) 201

Bacchetta, Bomhof, Pijlman, Mulders, PRD 72 (2005) 034030

Bomhof, Mulders, Pijlman, EPJ; hep-ph/0601171

Eguchi, Koike, Tanaka, hep-ph/0604003

Ji, Qiu, Vogelsang, Yuan, hep-ph/0604023

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